Periodic Juggling

Definition:

A period-n juggling pattern is a bijection such that for all .

If df is of period n then it might also have a period m for some divisor m of n. If n is the smallest period of df then any other period is a multiple of n; in this case we will say that f is a pattern of exact period n.

A period-n juggling pattern can be described by giving the finite sequence of non-negative integers for . Thus the pattern 51414 denotes a period-5 pattern; by Theorem 1 it is a 3-ball pattern since the ``period average'' of the height function is 3.

Which finite sequences correspond to juggling patterns? Certainly a necessary condition is that the average must be an integer. However this isn't sufficient. The sequence 354 has average 3 but does not correspond to a juggling pattern---if you try to draw an arrow diagram for a map f as above you'll find that no such map exists. This is also easy to see directly, for if and then

In order to find out which finite sequences represent juggling patterns we start by noting that a period-n pattern induces a permutation on the first n integers.

Lemma

If f is a period-n juggling pattern then

[Proof]

The Lemma implies that a juggling pattern f induces a well-defined injective, and hence bijective, mapping on the integers modulo n. Let denote the set and let denote the symmetric group consisting of all permutations (bijections) of the set . Then for every period n juggling pattern f there is a well-defined permutation that is defined by the condition

Theorem 2

A sequence of non-negative integers satisfies for some period-n juggling pattern f if and only if is a permutation of .

[Proof]

To see if 345 corresponds to a juggling pattern we add t to the t-th term and reduce modulo 3. The result is 021, which is a permutation, so 345 is indeed a juggling pattern (in fact a somewhat difficult one that is quite amusing). On the other hand, the sequence 354 leads, by the same process, to 000 which certainly isn't a permutation of .

• Remarks for Jugglers Only